Properties

Label 25215.f
Number of curves 8
Conductor 25215
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("25215.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25215.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25215.f1 25215h8 [1, 0, 0, -3630995, -2663397180] [2] 266240  
25215.f2 25215h6 [1, 0, 0, -226970, -41617125] [2, 2] 133120  
25215.f3 25215h7 [1, 0, 0, -184945, -57494170] [2] 266240  
25215.f4 25215h4 [1, 0, 0, -134515, 18977882] [2] 66560  
25215.f5 25215h3 [1, 0, 0, -16845, -390600] [2, 2] 66560  
25215.f6 25215h2 [1, 0, 0, -8440, 293567] [2, 2] 33280  
25215.f7 25215h1 [1, 0, 0, -35, 12840] [2] 16640 \(\Gamma_0(N)\)-optimal
25215.f8 25215h5 [1, 0, 0, 58800, -2917143] [2] 133120  

Rank

sage: E.rank()
 

The elliptic curves in class 25215.f have rank \(1\).

Modular form 25215.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} + 2q^{13} + q^{15} - q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.